`BookProof.ChapterAttentionSparse.one_sub_attendedMass_eq` (beta : ℝ) (s : Fin m → ℝ) (S : Finset (Fin m)) (i : Fin m) : 1 - attendedMass beta s S = ∑ l ∈ Sᶜ, scoreSoftmax beta s l

Proved

BookProof.ChapterAttentionSparse.one_sub_attendedMass_eq

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAttentionSparse.

BookProof.ChapterAttentionSparse.one_sub_attendedMass_eq (beta : ℝ) (s : Fin m → ℝ) (S : Finset (Fin m)) (i : Fin m) : 1 - attendedMass beta s S = ∑ l ∈ Sᶜ, scoreSoftmax beta s l

Formalization note: Lean 4 identifier BookProof.ChapterAttentionSparse.one_sub_attendedMass_eq.

Preamble

-- Generated from ChapterAttentionSparse.lean — theorem BookProof.ChapterAttentionSparse.one_sub_attendedMass_eq
import Definitions.Def_ChapterSoftmaxOrder
import Mathlib
import Definitions.Def_ChapterAttentionSparse
import Definitions.Def_ChapterSoftmaxSharpness
open BookProof.ChapterSoftmaxSharpness
open BookProof.ChapterAttentionSparse


open scoped BigOperators

noncomputable section


open BookProof.ChapterSoftmaxSharpness BookProof.ChapterSoftmaxOrder

variable {m : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]

Formal statement

theorem BookProof.ChapterAttentionSparse.one_sub_attendedMass_eq (beta : ℝ) (s : Fin m → ℝ) (S : Finset (Fin m)) (i : Fin m) :
    1 - attendedMass beta s S = ∑ l ∈ Sᶜ, scoreSoftmax beta s l := by sorry

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